Note

Click here to download the full example code

# Using Weights¶

One of the advantages of using a Green’s functions approach to interpolation is that we can easily weight the data to give each point more or less influence over the results. This is a good way to not let data points with large uncertainties bias the interpolation or the data decimation.

```
# The weights vary a lot so it's better to plot them using a logarithmic color scale
from matplotlib.colors import LogNorm
import matplotlib.pyplot as plt
import cartopy.crs as ccrs
import numpy as np
import verde as vd
```

We’ll use some sample GPS vertical ground velocity which has some variable uncertainties associated with each data point. The data are loaded as a pandas.DataFrame:

```
data = vd.datasets.fetch_california_gps()
print(data.head())
```

Out:

```
latitude longitude height ... std_north std_east std_up
0 34.116409 242.906804 762.11978 ... 0.0002 0.00037 0.00053
1 34.116409 242.906804 762.10883 ... 0.0002 0.00037 0.00053
2 34.116409 242.906805 762.09364 ... 0.0002 0.00037 0.00053
3 34.116409 242.906805 762.09073 ... 0.0002 0.00037 0.00053
4 34.116409 242.906805 762.07699 ... 0.0002 0.00037 0.00053
[5 rows x 9 columns]
```

Let’s plot our data using Cartopy to see what the vertical velocities and their uncertainties look like. We’ll make a function for this so we can reuse it later on.

```
def plot_data(coordinates, velocity, weights, title_data, title_weights):
"Make two maps of our data, one with the data and one with the weights/uncertainty"
fig, axes = plt.subplots(
1, 2, figsize=(9.5, 7), subplot_kw=dict(projection=ccrs.Mercator())
)
crs = ccrs.PlateCarree()
ax = axes[0]
ax.set_title(title_data)
maxabs = vd.maxabs(velocity)
pc = ax.scatter(
*coordinates,
c=velocity,
s=30,
cmap="seismic",
vmin=-maxabs,
vmax=maxabs,
transform=crs,
)
plt.colorbar(pc, ax=ax, orientation="horizontal", pad=0.05).set_label("m/yr")
vd.datasets.setup_california_gps_map(ax)
ax = axes[1]
ax.set_title(title_weights)
pc = ax.scatter(
*coordinates, c=weights, s=30, cmap="magma", transform=crs, norm=LogNorm()
)
plt.colorbar(pc, ax=ax, orientation="horizontal", pad=0.05)
vd.datasets.setup_california_gps_map(ax)
plt.tight_layout()
plt.show()
# Plot the data and the uncertainties
plot_data(
(data.longitude, data.latitude),
data.velocity_up,
data.std_up,
"Vertical GPS velocity",
"Uncertainty (m/yr)",
)
```

Out:

```
/home/travis/build/fatiando/verde/tutorials/weights.py:56: UserWarning: Tight layout not applied. The left and right margins cannot be made large enough to accommodate all axes decorations.
plt.tight_layout()
```

## Weights in data decimation¶

`BlockReduce`

can’t output weights for each data point because it
doesn’t know which reduction operation it’s using. If you want to do a weighted
interpolation, like `verde.Spline`

, `BlockReduce`

won’t propagate
the weights to the interpolation function. If your data are relatively smooth, you can
use `verde.BlockMean`

instead to decimated data and produce weights. It can
calculate different kinds of weights, depending on configuration options and what you
give it as input.

Let’s explore all of the possibilities.

```
mean = vd.BlockMean(spacing=15 / 60)
print(mean)
```

Out:

```
BlockMean(adjust='spacing', center_coordinates=False, drop_coords=True,
region=None, shape=None, spacing=0.25, uncertainty=False)
```

### Option 1: No input weights¶

In this case, we’ll get a standard mean and the output weights will be 1 over the variance of the data in each block:

in which \(N\) is the number of data points in the block, \(d_i\) are the data values in the block, and the output values for the block are the mean data \(\bar{d}\) and the weight \(w\).

Notice that data points that are more uncertain don’t necessarily have smaller weights. Instead, the blocks that contain data with sharper variations end up having smaller weights, like the data points in the south.

```
coordinates, velocity, weights = mean.filter(
coordinates=(data.longitude, data.latitude), data=data.velocity_up
)
plot_data(
coordinates,
velocity,
weights,
"Mean vertical GPS velocity",
"Weights based on data variance",
)
```

### Option 3: Input weights are 1 over the data uncertainty squared¶

If input weights are 1 over the data uncertainty squared, we can use uncertainty
propagation to calculate the uncertainty of the weighted mean and use it to define our
output weights. Use option `uncertainty=True`

to tell `BlockMean`

to
calculate weights based on the propagated uncertainty of the data. The output weights
will be 1 over the propagated uncertainty squared. In this case, the **input weights
must not be normalized**. This is preferable if you know the uncertainty of the data.

in which \(\sigma_i\) are the input data uncertainties in the block and \(\sigma_{\bar{d}^*}\) is the propagated uncertainty of the weighted mean in the block.

Notice that in this case the output weights reflect the input data uncertainties. Less weight is given to the data points that had larger uncertainties from the start.

```
# Configure BlockMean to assume that the input weights are 1/uncertainty**2
mean = vd.BlockMean(spacing=15 / 60, uncertainty=True)
coordinates, velocity, weights = mean.filter(
coordinates=(data.longitude, data.latitude),
data=data.velocity_up,
weights=data.weights,
)
plot_data(
coordinates,
velocity,
weights,
"Weighted mean vertical GPS velocity",
"Weights based on data uncertainty",
)
```

Note

Output weights are always normalized to the ]0, 1] range. See
`verde.variance_to_weights`

.

## Interpolation with weights¶

The Green’s functions based interpolation classes in Verde, like
`Spline`

, can take input weights if you want to give less importance to
some data points. In our case, the points with larger uncertainties shouldn’t have the
same influence in our gridded solution as the points with lower uncertainties.

Let’s setup a projection to grid our geographic data using the Cartesian spline gridder.

```
import pyproj
projection = pyproj.Proj(proj="merc", lat_ts=data.latitude.mean())
proj_coords = projection(data.longitude.values, data.latitude.values)
region = vd.get_region(coordinates)
spacing = 5 / 60
```

Now we can grid our data using a weighted spline. We’ll use the block mean results with uncertainty based weights.

Note that the weighted spline solution will only work on a non-exact interpolation. So we’ll need to use some damping regularization or not use the data locations for the point forces. Here, we’ll apply a bit of damping.

```
spline = vd.Chain(
[
# Convert the spacing to meters because Spline is a Cartesian gridder
("mean", vd.BlockMean(spacing=spacing * 111e3, uncertainty=True)),
("spline", vd.Spline(damping=1e-10)),
]
).fit(proj_coords, data.velocity_up, data.weights)
grid = spline.grid(
region=region,
spacing=spacing,
projection=projection,
dims=["latitude", "longitude"],
data_names=["velocity"],
)
```

Calculate an unweighted spline as well for comparison.

```
spline_unweighted = vd.Chain(
[
("mean", vd.BlockReduce(np.mean, spacing=spacing * 111e3)),
("spline", vd.Spline()),
]
).fit(proj_coords, data.velocity_up)
grid_unweighted = spline_unweighted.grid(
region=region,
spacing=spacing,
projection=projection,
dims=["latitude", "longitude"],
data_names=["velocity"],
)
```

Finally, plot the weighted and unweighted grids side by side.

```
fig, axes = plt.subplots(
1, 2, figsize=(9.5, 7), subplot_kw=dict(projection=ccrs.Mercator())
)
crs = ccrs.PlateCarree()
ax = axes[0]
ax.set_title("Spline interpolation with weights")
maxabs = vd.maxabs(data.velocity_up)
pc = grid.velocity.plot.pcolormesh(
ax=ax,
cmap="seismic",
vmin=-maxabs,
vmax=maxabs,
transform=crs,
add_colorbar=False,
add_labels=False,
)
plt.colorbar(pc, ax=ax, orientation="horizontal", pad=0.05).set_label("m/yr")
ax.plot(data.longitude, data.latitude, ".k", markersize=0.1, transform=crs)
ax.coastlines()
vd.datasets.setup_california_gps_map(ax)
ax = axes[1]
ax.set_title("Spline interpolation without weights")
pc = grid_unweighted.velocity.plot.pcolormesh(
ax=ax,
cmap="seismic",
vmin=-maxabs,
vmax=maxabs,
transform=crs,
add_colorbar=False,
add_labels=False,
)
plt.colorbar(pc, ax=ax, orientation="horizontal", pad=0.05).set_label("m/yr")
ax.plot(data.longitude, data.latitude, ".k", markersize=0.1, transform=crs)
ax.coastlines()
vd.datasets.setup_california_gps_map(ax)
plt.tight_layout()
plt.show()
```

Out:

```
/home/travis/build/fatiando/verde/tutorials/weights.py:303: UserWarning: Tight layout not applied. The left and right margins cannot be made large enough to accommodate all axes decorations.
plt.tight_layout()
```

**Total running time of the script:** ( 0 minutes 4.221 seconds)